"But this is not mathematics!"-mathematicians and secondary teachers explore the affordances of tertiary mathematics for teaching secondary probability

Tertiary mathematics has a central place in teacher education, yet in recent years there is growing evidence that realizing its potential affordances in secondary mathematics teaching is far from trivial. Research suggests that utilizing tertiary mathematics in secondary teaching requires interweaving it with knowledge for teaching secondary mathematics. Little is known about the underlying processes, which are often tacit and highly personal. In this article we analyze affordances of tertiary mathematics for teaching secondary probability. A group of mathematicians and experienced secondary teachers jointly inquired into the mathematics that could be addressed in school when discussing a popular probability game - the River Crossing game (henceforth "the game"). This context was chosen as an extreme case, in the sense that the mathematics underlying the game is so nuanced and complex that applying tertiary knowledge to mathematize and understand it is generally not feasible for secondary teachers. Thus, it is not clear how tertiary mathematics can inform teachers about using the game in class. Our analysis shows how the conflicting perspectives of teachers and mathematicians on what mathematics students may learn by playing the game initially hindered utilization of tertiary mathematics. Nevertheless, rapprochement was achieved, highlighting four different trajectories for interweaving knowledge of tertiary mathematics with knowledge for teaching secondary mathematics towards using the game in ingenious ways that respect both mathematical and pedagogical concerns. Our findings suggest that tertiary mathematics may have affordances for secondary mathematics teaching even in situations where teachers lack tertiary-level understanding of the underlying subject-matter.